Abstract
The inflation G I of a graph G with n ( G ) vertices and m ( G ) edges is obtained from G by replacing every vertex of degree d of G by a clique K d . A set S of vertices in a graph G is a paired dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired domination number γ p ( G ) is the minimum cardinality of a paired dominating set of G . In this paper, we show that if a graph G has a minimum degree δ ( G )⩾2, then n ( G )⩽ γ p ( G I )⩽4 m ( G )/[ δ ( G )+1], and the equality γ p (G I ) = n(G) holds if and only if G has a perfect matching. In addition, we present a linear time algorithm to compute a minimum paired-dominating set for an inflation tree.
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